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Mathematisches Forschungsinstitut Oberwolfach Geometric Knot Theory

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Mathematisches Forschungsinstitut Oberwolfach Geometric Knot Theory Mathematisches Forschungsinstitut Oberwolfach Report No. 22/2013 DOI: 10.4171/OWR/2013/22 Geometric Knot Theory Organised by Dorothy Buck, London Jason Cantarella, Athens John M. Sullivan, Berlin Heiko von der Mosel, Aachen 28 April – 4 May 2013 Abstract. Geometric knot theory studies relations between geometric prop- erties of a space curve and the knot type it represents. As examples, knotted curves have quadrisecant lines, and have more distortion and more total cur- vature than (some) unknotted curves. Geometric energies for space curves – like the M¨obius energy, ropelength and various regularizations – can be minimized within a given knot type to give an optimal shape for the knot. Increasing interest in this area over the past decade is partly due to various applications, for instance to random knots and polymers, to topological fluid dynamics and to the molecular biology of DNA. This workshop focused on the mathematics behind these applications, drawing on techniques from algebraic topology, differential geometry, integral geometry, geometric measure theory, calculus of variations, nonlinear optimization and harmonic analysis. Mathematics Subject Classification (2010): 57M, 53A, 49Q. Introduction by the Organisers The workshop Geometric Knot Theory had 23 participants from seven countries; half of the participants were from North America. Besides the fifteen main lectures, the program included an evening session on open problems. One morning session was held jointly with the parallel workshop “Progress in Surface Theory” and included talks by Joel Hass and Andre Neves. Classically, knot theory uses topological methods to classify knot and link types, for instance by considering their complements in the three-sphere. But in recent decades there has been increasing interest in connecting these topological knot types with geometric properties of the various space curves representing them, and 1314 Oberwolfach Report 22/2013 in finding optimal geometric shapes for a given knot type by minimizing various energy functionals. To find an optimal shape for a given knot, one usually minimizes some geometric energy. We emphasize that the concept of “energy” is very general here. For instance, we can build polygonal “stick knots” of fixed or varying edge length and let the energy be the number of sticks; we could also restrict these sticks to lie in a given grid or to form an arc presentation in an open book. Average crossing number and total curvature are classical examples of energy functions, for which the infimal value is not achieved but instead recovers diagrammatic information (minimum crossing number and bridge number, respectively). It has long been hoped that following the gradient flow of a more suitable energy functional for curves would lead to an effective method for simplifying the shape of knots. For example, one might load a curve with electric charge and let its shape evolve by electrostatic repulsion. A scale-invariant version of the Coulomb potential, properly renormalized, was introduced by O’Hara and shown by Freedman, He and Wang to be M¨obius-invariant. This gives a very useful energy for knots and links, where many open problems remain. Another natural question asks how much rope is needed to tie a given knot or link. Mathematically the rope is idealized to have fixed circular cross-section: it is a normal tube around some core curve. We minimize the ropelength, the scale- invariant quotient of length over thickness. Here thickness (or reach in Federer’s terminology) is the maximal radius of a normal tube and has many equivalent formulations useful in different contexts. For instance it is bounded both by the local radius of curvature and by the closest approach of two different strands of the knot. On the other hand, it is also the maximal Menger curvature of all triples of points on the knot, that is, the infimal radius of circles intersecting the knot three times. p Replacing the maximum of Menger curvature (an L∞ norm) by various L norms, one obtains a whole family of knot energies, interpolating between thickness on the one hand and repulsive Coulomb-type potentials on the other. When these energies are finite (or minimized) for a given curve, the smoothness of the curve and its geometric complexity are both controlled. Recently, the spaces of finite energy curves for these and related energies have been characterized in terms of fractional Sobolev spaces. A similar characterization for the M¨obius energy made it possible to analyze its gradient flow, and there is hope that these techniques can be extended to study geometric flows for integral Menger curvature as well. In the limit, this could also lead to an analytic understanding of the gradient flow for ropelength. Moreover, intricate bootstrapping arguments for fractional orders of differentiability were recently developed to prove smoothness of critical points for several of these energies. In addition, integral Menger curvature and its interpolating relatives have successfully been generalized to submanifolds of arbitrary dimension and co-dimension in Euclidean space, which opens up the search for corresponding results in the context of higher dimensional (knotted) Geometric Knot Theory 1315 submanifolds, such as finiteness of isotopy types with bounds on integral Menger curvature and regularity theorems for finite or minimal energy submanifolds. Another set of interesting questions arises from considering the average values of geometric functionals over spaces of curves instead of the minimum values. For instance, we might ask for the average total curvature of a class of curves, or the average knot type. To get sensible answers here, we must restrict our attention to finite-dimensional spaces of curves such as the space of polygons with a given length and a given number of edges. We must also choose a natural probability measure to integrate our functionals against; this measure also provides the setting for a theory of random knots. Such random knotted polygons are thought to provide good models for knotted polymer molecules. The relationship between the averages over random polygons in a given knot type to the minimum values for this knot type (and to the topology) is an area of active research. Progress here could lead to important advances in the statistical physics of polymers and other entangled systems. The fifteen talks and the open problem session are documented in the remainder of this report. The workshop schedule also left free time for informal mathematical interactions, and many fruitful discussions developed. On the lighter side, Colin Adams directed an evening of mathematical theater based on some of his humorous columns for the Mathematical Intelligencer. More than half the workshop participants acted in one or more of the skits, and the appreciative audience included almost everyone from both workshops. In addition to the traditional Wednesday hike to St. Roman, many participants walked to the Museum for Minerals and Mathematics after the last talk on Friday afternoon. Geometric Knot Theory 1317 Workshop: Geometric Knot Theory Table of Contents Colin Adams (joint with Thomas Crawford, Benjamin DeMeo, Michael Landry, Alex Tong Lin, MurphyKate Montee, Seojung Park, Saraswathi Venkatesh, Farrah Yhee) Turning knots into flowers ........................................1319 Simon Blatt Gradient flow for the M¨obius energy ...............................1320 Ryan Budney Universal constructions on spaces of knots with relations to finite-type invariants ................................................... ...1323 Yuanan Diao On the ropelength problem of knots ................................1325 Claus Ernst (joint with Y. Diao, E. Rawdon, U. Ziegler) The effect of confinement on knotting and geometry of random polygons 1329 Joel Hass (joint with Alexander Coward ) Distinguishing physical and mathematical knots and links .............1332 Gyo Taek Jin (joint with Hyuntae Kim) Tabulation of prime knots by arc index .............................1333 Gyo Taek Jin (joint with Hwa Jeong Lee) Arc index of pretzel knots of type (−p,q,r) ........................1334 S lawomir Kolasi´nski Menger-type curvature in higher dimensions ........................1335 Kenneth C. Millett (joint with Jorge A. Calvo, Akos Dobay, Laura Plunkett, Eric Rawdon, Andrzej Stasiak) Average geometric and topological properties of open and closed equilateral polygonal chains .................................................1337 Andre Neves (joint with Ian Agol and Fernando Marques) Min-max theory and the energy of links ............................1339 Jun O’Hara Three topics in knot energies ......................................1340 Eric J. Rawdon (joint with Kenneth C. Millett, Joanna I. Su lkowska, Andrzej Stasiak) Knotted arcs in open chains, closed chains, and proteins ..............1343 1318 Oberwolfach Report 22/2013 Philipp Reiter (joint with Simon Blatt) Regularity theory for knot energies .................................1344 Clayton Shonkwiler (joint with Jason Cantarella) The symplectic geometry of polygon space ...........................1347 Pawe lStrzelecki (joint with Heiko von der Mosel and Marta Szuma´nska) Menger curvature as a knot energy ................................1350 Minutes of the Open Problem Session ..............................1353 Geometric Knot Theory 1319 Abstracts Turning knots into flowers Colin Adams (joint work with Thomas Crawford, Benjamin DeMeo, Michael Landry, Alex Tong Lin, MurphyKate Montee, Seojung Park, Saraswathi Venkatesh, Farrah Yhee)
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